Abstract
Keywords
1 Introduction
Polaritons in two-dimensional (2D) semiconductor materials (e.g., plasmon-polaritons in graphene1,2 or exciton–trion polaritons in transition metal dichalcogenides3,4) have attracted significant interest as a promising platform for developing novel optoelectronic devices because of the variety of tools for controlling the density of carriers. Photocarrier injection via high-intensity pulses is useful to probe ultrafast transient responses of such polaritons,5
Owing to quantum capacitance effect,12 spatially modulated carrier densities under electrochemical equilibrium give rise to superlattice (SL) electric potentials for Dirac electrons, given as
Figure 1.Engineering of SL electric potential in graphene. (a) Physical realization: field-effect carrier density modulation using a metagate/backgate combination. Inset: SL potential
Under a periodic SL potential, the conical Dirac dispersion deforms into subbands.15
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In this study, we show that a one-dimensional (1D) SL potential in a 2D Dirac electron gas leads to coherent intersubband resonant features of the optical conductivity corresponding to the emergence of a new electrically controlled quasi-particle: a hybrid intersubband-plasmon-polariton (HIPP). Resonant ISBTs occur between flat subbands in the electronic structure that are generated by potential barriers separating the adjacent SL periods. When electron tunneling is sufficiently weak, multiple electronic states that are quasi-confined to individual SL periods can form, giving rise to a ladder-like equispaced energy level. Transitions between these subbands are responsible for the ultrastrong resonant enhancement of the oscillator strengths of the ISBT at quantized electrically controlled frequencies. We show that the HIPP dispersion relations—propagation wavenumber and decay rate as a function of their frequency —can be systemically controlled via the combination of a patterned metagate and a continuous backgate, as shown in Fig. 1(a). The metagate can be used to control the underlying plasmon-polariton dispersion, whereas the backgate controls the ISBT frequencies and the Rabi-splitting strengths. Our results demonstrate that SL engineering in 2D materials—a versatile technique that has been used to study exotic electronic transport properties—can also be used for creating novel on-demand polaritonic materials.
2 Results
Figure 1(a) depicts the geometry of the setup for realizing electrically controlled HIPPs. The combination of a patterned (metagate)21 and unpatterned (backgate) electrical gates is used to attain better tunability of the free-carrier density in graphene.10,19 The function of the backgate placed underneath the metagate is to define the carrier densities in those regions of graphene that are not screened by the metagate. If the duty cycle of the metagate air gaps is very small (), the baseline Drude weight of graphene’s optical conductivity (proportional to the local chemical potential ) is primarily defined by the metagate voltage . On the other hand, the modulation depth of the SL potential is tuned with the backgate voltage . The highly efficient screening of the backgate by the metagate is illustrated by the results of an electrostatic simulation of Eq. (1) shown in Fig. 1(b), where we observe that very few of the electric field lines emanating from the backgate are reaching graphene. As a result, the doped carrier density in the region above the metallic grating is flat and controlled by , whereas the region above the air gaps can be nearly depleted when sufficiently negative voltage is applied to the backgate. The resulting shape of shown in Fig. 1(c) can be viewed as a periodic array of wide-square potential wells separated by narrower potential barriers with the heights directly controlled by .
The electronic subband structures for Dirac electrons in 1D SL potentials periodic along the axis are calculated by solving the Dirac Hamiltonian equation with the Bloch ansatz :
For vanishing , the conical linear dispersion is exactly preserved, even in the presence of strongly modulated [Fig. 2(a) left] due to Klein tunneling.22,23 At sufficiently large , in contrast, there occur several flat subbands around and below the Fermi surface [Fig. 2(a) right], as the bound states are formed inside the potential wells via the total internal reflection of electrons with Dirac dispersion (TIREDD). As clearly shown in Fig. 2(b), these bound states are confined within a finite region inside each SL period, with their wave functions evanescently decaying across the potential barriers separating adjacent SL periods. Therefore, Dirac electrons are guided along the axis with negligible tunneling across potential barriers. The negligible coupling between the bound states in the adjacent SL periods thus manifests as flat subbands in the band diagrams.
Figure 2.The TIREDD mechanism is responsible for flat subbands of graphene electrons in SL electric potentials. (a) Left: no intersubband gaps for
The TIREDD condition is satisfied when a Dirac electron with energy and is incident from the -oscillatory domain—the electron waveguide (EW)—onto the -evanescence domain—the electron cladding (EC). This scenario is best visualized by plotting the local iso-energy contours at the centers of the EW and EC domains. Inside the EW domain, those are represented by a circle in the momentum space: , where and . On the other hand, inside the EC domain, and the electron wave function exponentially drops along the direction with the evanescent decay rate . Therefore, the EW and EC domains are defined as the regions in where [] conditions are, respectively, satisfied.
Notably, the EC domains do not exist for the electrons with , leading to Klein tunneling. Also, as illustrated in Fig. 2(c), TIREDD occurs not only when an electron travels from a lower-potential to a higher-potential domain—a more familiar picture—but it also in the opposite case where the electron travels from a higher potential domain to a lower potential domain. The latter case corresponds to an antiparticle in the high-energy physics context, or to a valence band electron in the graphene context. As a result, bound states out of valence band electrons are formed around the potential barriers [see state (iii) in Fig. 2(b)]. Each flat subband in the band diagram in Fig. 2(a) corresponds to a TIREDD-based bound state, and the flatness depends on whether the width of the ECs satisfying is sufficiently thicker than the decay length . We note that the mean free path of electrons in hBN-encapsulated graphene can be as high as at the room temperature and near at cryogenic temperatures .24 Thus we expect that the TIREDD condition can be realized in realistic experiments because the mean free path in high-quality graphene samples is much longer than the width of the SL potential wells.
Another notable feature of the TIREDD-based flat subbands is the ladder-like energy level spacing around the Fermi surface. With a rough approximation (for more detailed analysis, see Supplementary Material), the bound-state energy levels of Dirac electrons in a square potential well (for now, let us consider the conduction band TIREDD only) are given as , where is the width of potential well, is the potential inside the well, and is the band index. Even though this expression is not completely linear in , it quickly approaches the asymptotic linear relation when is only twice greater than . Thus the ISBT energy is maintained as nearly uniform over a substantial portion of the Fermi surface (see Fig. 3), which resonantly enhances the oscillator strength of the ISBT at certain quantized frequencies given as integer multiples of . In Fig. 3, six or seven bands are altogether contributing to the ISBT nearly at the same frequency.
Figure 3.The massless dispersion of Dirac electrons makes the ISBT energy be nearly uniform over a broad region of the Fermi surface (
Ultrastrong ISBT optical responses manifest as resonant features in graphene optical conductivity, which is calculated by the Kubo formula under random phase approximation (RPA),20,25 given as
Figure 4.Optical conductivity and ISBT resonances for SL-modulated graphene electrons. (a) Real part of the conductivity
Figure 5 shows the resulting HIPP dispersions featuring the ultrastrong coupling (USC) between the ISBT and the underlying plasmon polaritons. If the system is spatially homogeneous, the polariton eigenmodes in graphene appear as the zeros of the scalar dynamical dielectric function .25,29 Here is the dynamical capacitance of the system,30 which connects the dynamic carrier density oscillation and the dynamic electric potential field on graphene : . In essence, the dynamic capacitance encodes the information about the dielectric environment around graphene, in contrast to the conductivity that encodes the dielectric property of graphene itself. With a periodic modulation along the axis like in our system, the dynamical dielectric function is given as a matrix form,
Figure 5.HIPP dispersion with USC and far-field detection of HIPPs. (a) Density of states or
With a moderate depth of the SL modulation ( case), the HIPP dispersion is similar to the plasmon-polariton dispersion with no modulation, and the ISBT feature is very subtle. As the SL modulation gets deeper, there emerge several HIPP branches resulting from the hybridization between the underlying plasmon polaritons and the ISBT resonances. With an extreme modulation ( case), we observe a huge Rabi splitting () between the lowest branch and the second lowest, which is even comparable to the ISBT frequency itself (). In such a USC regime, a recent study reported that the electronic band structure of the material could be modified in return, due to the vacuum fluctuation of the strongly interacting polaritonic modes.31 We believe that our system would exhibit a similar behavior, but we did not consider such additional corrections in this work. Figure 5(d) illustrates that this HIPP phenomenon can be detected in the far-field reflection as well. Each peak in the reflection spectra corresponds to the mode along the second-lowest HIPP branch. In the reflection calculation, we assumed that the backgate substrate is silicon doped with a carrier density of . We note that the depth of reflectivity peaks (or dips) can be engineered (up to an order of magnitude) with the oxide substrate thickness and the silicon backgate doping density (see Supplementary Material).
3 Discussion
The emergent HIPPs found in our proposed system have several unique features compared to the usual intersubband-polaritons or intersubband-plasmon-polaritons studied in other platforms. First, the quantum well structure is given along the direction of the polariton propagation. Accordingly, the ISBT of our system occurs through in-plane electric fields along the axis, which allows the far-field detection even with normal incidence of light. In contrast, conventional ISBT structures are based on vertical engineering of quantum wells,31
Third, as discussed earlier, the linear dispersion of Dirac electrons allows multiple (6 to 7) bands with equispaced energy levels to resonantly build up the ISBT strength. This resonant enhancement from multiple ladder-like bands near the Fermi surface still appears in massive electron systems (see the Supplementary Material), as the potential well has a finite depth with slanted sidewalls and the quadratic dispersion in momentum gets linearized at the vicinity of the Fermi surface. Our analysis in the Supplementary Material reveals that, given the same SL potentials shown in Fig. 2(b), it requires an extremely tiny effective mass ( of an electron mass ) to obtain the same ISBT energy level spacing found in graphene. Since the ISBT energy level spacing in massive electron systems scales as inversely proportional to the square root of the effective mass, too high effective masses would require much narrower lateral quantum wells to maintain the same ISBT energy level spacing (or much better sample mobility and reduced Drude loss to maintain the same HIPP quality factor). Thus we suggest that 2D electron gas at InGaAs/GaAs interfaces, where the effective mass of electrons is 4% to 7% of ,34 would be a reasonable material platform to study this HIPP phenomenon with massive electrons. In this paper, however, we mainly focused on the massless Dirac electrons, as metagate-tuned SL potential and band structure engineering have been readily implemented in graphene, whereas such techniques have not been attempted or matured with 2D electron gases in GaAs material systems, to the best of our knowledge.
We emphasize again that the HIPPs shown in Fig. 5 operate in the ultrastrongly coupled regime, featuring a giant Rabi splitting that becomes comparable to the ISBT frequency. In this regime, several quantum electrodynamic phenomena can arise, such as material bandgap renormalization31 or antiresonant coupling that breaks the rotating wave approximation and the Kubo conductivity formula.35 Therefore, more precise determination of the HIPP dispersion would require a full quantum description of the USC physics, which we leave as a future work. Finally, even apart from the HIPP physics, the TIREDD-induced ladder-like energy bands themselves can be useful for high-harmonic generations. Nonlinear optical responses can be resonantly enhanced by engineering the equispaced energy level of subbands in quantum well structures.36 Thus the 1D SL potential in graphene can also be used as a novel material platform for nonlinear optics.
In conclusion, our study suggests that the SL engineering in 2D materials can lead to the discovery of novel polariton phenomena emerging from the deformed electronic band structures. The modified subband structure of Dirac electrons under a 1D SL adds a completely new dimension to the polariton composition, leading to the formation of the HIPPs. This emergent HIPP is easily tunable by the double-gating scheme, provides a way to detect the SL-induced band structure changes with a far-field optical measurement, and becomes suitable for the study of quantum and nonlinear optics based on ultrastrong light–matter interaction. Introducing a 2D SL18 or patterning other 2D materials beyond graphene37,38 could lead to more opportunities to study emergent polaritons with other novel formation mechanisms.
We envision that the experimental implementation of the reported phenomenon is readily realizable, considering the state-of-the-art fabrication schemes of metagate-based graphene SLs11,17,19,30 and availability of ultrahigh quality of graphene samples.24,27 The polaritonic dispersions can be directly probed by near-field scanning optical microscopies1,2,5,7,9,11,27,28,30 or can be probed in a rather simple experimental setup through far-field reflections, as suggested in this work. The speed of polariton modulation through ultrafast electronics can extend to tens of gigahertz, limited by the external electronic settings (e.g., network analyzer or RF probes) rather than the electron relaxation/response time in graphene.39
4 Materials and Methods
SL potentials and the electrostatic field lines shown in Fig. 1 were computed by COMSOL Multiphysics with a nonlinear solver, given the nonlinear boundary condition set by Eq. (1), by setting the graphene sheet as a surface charge density element.40 Electronic band structures of Dirac electrons under SL potentials in Figs. 2 and 3, HIPP optical conductivity in Fig. 4, and polaritonic density of states in Fig. 5 were computed by an in-house (rigorous coupled-wave analysis) code based on Eqs. (2)–(4). Detailed high-level descriptions of the numerical methods are provided in the Supplementary Material.
Minwoo Jung received his BA, MA, and PhD degrees in physics from Cornell University in 2017, 2020, and 2022, respectively. His research mainly focuses on utilizing nanopolaritons in two-dimensional materials as platforms for advanced photonics.
Gennady Shvets received his PhD in physics from MIT in 1995. He is a professor of applied and engineering physics at Cornell University. His research interests include nanophotonics, optical and microwave metamaterials and their applications (including biosensing, optoelectronic devices, and vacuum electronics), topological concepts in photonics, and ultraintense laser–matter interactions. He is the author or co-author of more than 200 papers in refereed journals. He is a fellow of the American Physical Society, Optica, and SPIE.
References
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[11] L. Xiong et al. Programmable Bloch polaritons in graphene. Sci. Adv., 7, eabe8087(2021).
[27] G. X. Ni et al. Fundamental limits to graphene plasmonics. Nature, 577, 530-533(2018).
[30] L. Xiong et al. Photonic crystal for graphene plasmons. Nat. Commun., 10, 4780(2019).
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